Statistical analysis has addressed the problem of estimating the effects of various conditions on different types of outcomes. As discussed below, in the case of a categorical outcome, the prior art has not produced a concise and informative display of these effects.
The simple linear regression model fits a straight line to a continuous outcome as a function of some predictor variable. The graphical display of the straight line makes obvious the statistical effect of the predictor on the outcome.
When the outcome is not continuous but contains say J categories, there is a multiplicity issue in that there are J-1 possible effects of a continuous predictor variable on the outcome, or, for a categorical predictor containing I categories or conditions, as many as (I-1)*(J-1) possible effects. See, e.g., Clogg and Shihadeh (1994), which is hereby incorporated herein by reference, at page 17. The potentially large number of effects makes it difficult to create a concise yet informative graphical display for this situation. Moreover, there are several competing statistical models for describing effects on categorical outcomes.
Duncan & McRae (1979) described an important class of models for a categorical outcome and presented some graphical displays for these models. However, these displays show odds and odds ratios pertaining only to the predictor variables and do not relate to the effects of the predictor variables on the outcome variable(s), and hence have no appeal as a display of effects.
Goodman (1991), which is hereby incorporated herein by reference, summarized the current state of development for these and similar models for two categorical cross-classified variables (i.e., a row variable and a column variable), and suggested certain models for more than two variables. Some of the models impose a fixed spacing between the rows (and/or columns) while others include "spacing parameters" which allow the spacing to be estimated from the data. Goodman also commented on a plot of certain transformations of observed data that can assist the statistician in determining which type of model may provide a good fit.
In addition to the multiplicity issue mentioned above, Goodman (1991) pointed out that none of the effects from these models are unique in that they are subject to certain identifying standardizations. He illustrated the use of two such standardizations. The first, which he called the unweighted average standardization, assigns equal weight to each row and also weights each column equally. The second, which he called the weighted average standardization weights each row proportionate to the total frequency count for that row and weights each column proportionate to the total frequency count for that column.
Since the important class of models discussed above are relatively new and continue to evolve, only a few of these models can be estimated at all with current computer programs. For those models that can be estimated by current computer programs, the use of the programs is complex and the interpretation of results is especially difficult. In a new book on these models, it is pointed out (Ishii-Kuntz, 1994, p. 11) that for the few models that can be currently estimated, not only do the current computer programs not display interpretable effects (i.e., "odds ratios") graphically, but most do not output them at all. Rather, they output in tabular form logarithmic transformations of the interpretable effects.
An important goal of researchers who analyze a categorical outcome is to use statistical models to identify and describe effects in their data and to obtain a clear interpretation of these effects. The goal of fitting a model to categorical outcome data and obtaining a useful display of the effects is complicated by the general multiplicity issue, the availability of a wide range of possible models, and the necessity for using an unweighted or weighted standardization or some other kind of standardizing restriction(s) to identify and define these effects. Accordingly, there is a need for a concise and informative display of the effects of various conditions (i.e., predictors) on a categorical outcome.